Schatten class weighted composition operators on Bergman spaces of bounded strongly pseudoconvex domains
Let \(\Omega\) be a bounded strongly pseudoconvex domain in \(\mathbb{C}^n\). Let \(u\) be an analytic function on \(\Omega\) and \(\varphi\) be an analytic self-map of \(\Omega\). The weighted composition operator \(uC_\varphi\) is defined as follows: for an analytic function \(f\) on \(\Omega\), define \(({uC_\varphi})f(z) = u(z)f({\varphi (z)})\). Recall that an analytic function \(f \in A^2 (\Omega)\) if and only if \(\left\| f \right\|^2 = \int_\Omega {\left| {f(z)} \right|^2 \,dv(z)} < \infty\), where \({dv}\) denotes the Lebesgue volume measure on \(\mathbb{R}^{2n}\). For a finite complex Borel measure \(\mu\) on \(\Omega\), the Berezin transform \(\tilde \mu (z)\) is defined by \(\tilde \mu (z) = \int_\Omega {\left| {k_z (w)} \right|^2 \,d\mu (w)}\), \(z \in \Omega \). Let \(\beta ({z,w})\) be the Kobayashi distance function on \(\Omega\). For any \( z \in \Omega\) and \(r > 0\), let \(B({z,r}) = \{ w \in \Omega :\beta ({z,w}) < r\}\). For a given finite complex Borel measure \(\mu\) on \(\Omega\), the function \(\hat \mu ^r (z)\) on \(\Omega\) is defined by \[ \hat \mu ^r (z) = \frac{{\mu ({B({z,r})})}}{{v({B({z,r})})}},\quad z \in \Omega. \] The essential norm, boundedness and compactness for \(uC_\varphi\) from Bergman spaces \(A^p (\Omega)\) to \(A^q (\Omega)\), \(1 < p \leq q < \infty\), were characterized in [\textit{Ž. Čučković} and \textit{R.-H. Zhao}, Math. Proc. Camb. Philos. Soc. 142, No. 3, 525--533 (2007; Zbl 1126.47024)]. Combining these results with the methods of [\textit{Ž. Čučković} and \textit{R.-H. Zhao}, J. Lond. Math. Soc., II. Ser. 70, No. 2, 499--511 (2004; Zbl 1069.47023)] and [\textit{K. Zhu}, J. Oper. Theory 20, No. 2, 329--357 (1988; Zbl 0676.47016)], the paper under review provides an alternative characterization of boundedness and compactness of weighted composition operators on \(A^2 (\Omega)\). Besides, motivated by the above references, the Schatten \(p\)-class weighted composition operators are also characterized as follows: Let \(u\) be an analytic function on \(\Omega\) and \(\varphi\) be an analytic self-map of \(\Omega\), furthermore, let \(p\geq2\). Let \( dv_u (z) = \left| {u(z)} \right|^2 \,dv(z)\) and \(\mu _u = v_u \circ \varphi ^{ - 1}\), which is the pull-back measure of \(v_u\) induced by \(\varphi\). Then the following are equivalent: {\parindent=8mm \begin{itemize}\item[(i)] \(uC_\varphi \in \mathcal{S}_p\). \item[(ii)] \(\int_\Omega {({\tilde \mu _u (z)})^{p/2} K({z,z})\,dv(z)} < \infty\). \item[(iii)] \(\int_\Omega {({\hat \mu _u^r (z)})^{p/2} K({z,z})\,dv(z)} < \infty\). \item[(iv)] \(\sum_{n = 1}^\infty {({\hat \mu _u^r ({a_n })})^{p/2} } < \infty ,\) where \(\{ a_n \}\) is some sequence in \(\Omega\) independent of \(\mu _u\). \end{itemize}}
scientific article; zbMATH DE number 6132915